The crucial point in man's mathematical history came when more than patterns were required; when more was needed than to look inside the cave to assure himself that both children were present, or a glance at his rack of stone axes to convince himself that all four spares were in place.
At some point, man found it necessary to communicate numbers. He had to go to a neighbor and say, "Listen, old man, you didn't lift one of my stone axes last time you were in my cave, did you?" Then, if the neighbor were to say, "Good heavens, what makes you think that?" it would be convenient to be able to say, "You see, friend, I had four spares before you came to visit and only three after you left."
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From Mathematical Experiences for Young Children, by Louise Binder Scott and Jewell Garner
Teachers have always known that children seem happier when involved in active learning and discovery . . .. While no one expects children to make new discoveries in an establish discipline like mathematics, one can assume that they will find ways of solving problems that are new to them. They cannot create new concepts, but they can and should discover many mathematical patterns and relationships for themselves.
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Vocabulary development is one of the essential ingredients in concept formation. Therefore, a mathematical readiness program necessarily requires language emphasis. . . .
Lewis suggests that some children are capable of successful transposition without verbal symbols of the process involved. * Yet as situations become more complex, children seem increasingly unable to deal with them without some verbal ability. A child may hold thoughts in mind without using words, but as relationships become more complex, verbalizing them is an invaluable aid to understanding.
Children must acquire the mental structure used in describing collections, in making comparisons, in drawing conclusions, and so forth. Eventually, they must learn to translate language into mathematical symbols and to interpret the mathematical code in the mother tongue. Since reasoning is so closely related with the ability to understand and use language, increasingly precise mathematical vocabulary must be acquired. Verbalization without understanding is likely to be detrimental to concept formation, for each level of a child's development depends upon previously acquired knowledge. Concepts become linked to one another in a growing structure of related ideas.
* M. M. Lewis, Language Thought and Personality (London: George G. Harrap, 1965).
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[On pages 14-15, a long section on asking questions as a way of helping children learn:
"What did you do?"
"How did it happen?"
"Why did you do this first? do this next?"
"If you had done anything else, would you still have the same answer?"]
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[Chapter 3: Beginning Geometry Concepts, opens with a discussion of developmental psychology (Oh, how I love this stuff!) as it relates to the child's growing perception of shape, relative size, distance, and perspective.]
Piaget's work with children shows that first concepts of space are topological. In topology, a triangle, square, and circle are equivalent; that is, one shape can be squeezed or stretched to form another.
Often a teacher says, "My children cannot seem to recognize shapes or follow directions pertaining to shapes." Some young children may lack visual perception and may require the tactile experiences of feeling two- or three-dimensional figures, tracing around templates, and following dots which result in simple configurations. Looking at a shape and being told that it is a square or a triangle are not sufficient. Many different kinds of activities, games, and discussions must be employed so that children will see the shape in their minds and the idea will become fixed. When children act upon an object physically, their mental images become stronger.
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The ability to seriate or place objects in order according to some criterion such as height is a conceptual development necessary for the acquisition of classification skills. Size is more easily perceived than class, and therefore the ability to order develops somewhat prior to the skill of sorting. Before objects can be classified, their relation to other objects must be recognized.
The idea of things belonging together because of certain characteristics is fundamental to the development of logical thinking. A child must be able to classify according to concrete, visible characteristics such as size, shape, color, and so forth, before the classification of sets according to the abstract property of number can be understood.
The class does not exist in isolation, but is usually part of a larger system. A robin belongs to a class of birds, but birds are included in the class of animals, and animals are included in the class of living things. Older children will be able to make a hierarchical classification based on this inclusion relation and from this activity will develop the understanding of the set-subset concept.
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A set is a collection of people, objects, ideas, and so forth, whose membership in the group can be determined. A given set is constant, conserved, and unchanged when members are rearranged or placed in different positions. Developing an understanding of set, how it relates to other sets and how to use sets, lays a foundation for more formal mathematical activities. The child's experimentation with sorting and classifying materials is followed by experiences with sets that help develop an understanding of addition and subtraction.
Ask: "What do we call a set of pups? a set of geese? a set of people who sing? a set of horses? a set of books? etc."
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[C]hildren usually conserve number by the age of seven years; they know that five pennies are still five pennies regardless of arrangement. Conservation of quantity develops somewhat later. A child who reasons logically in determining that number is constant may still believe that a given amount of liquid poured from one container into another of a different shape becomes less or more. Conservation of length, area, weight, and volume may not be operational until even later.
The various types of conservation do not occur at the same time in children's thinking. they do occur in the same order usually: first, conservation of number, then quantity, then weight (mass), and finally volume at around ten to eleven years of age.*[pp.87-88]
* Richard Copeland, How Children Learn Mathematics, 2nd ed. (New York: Macmillan Co., 1970), p. 95.
At about age seven years most children have acquired the ability to think back or reverse an action mentally. This concept of reversability grows out of actions children perform on objects in the world around them....
The concept of reversability is essential to the understanding of operations on number. For example, subtraction must be grasped as the inverse of addition; that is, if to 6 the child adds 3 and then subtracts 3, the solution is 6, which is the given number.
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It is essential that the difference between number and numeral be emphasized. Using the word number when referring to ideas and the word numeral when reading and writing names for the ideas will help individuals understand the distinction and will enable them to employ the terms correctly.
[On using the number line:]
It is important that the children realize that they should count units (steps or spaces) rather than the vertical marks separating the units during initial experiences with the number line. After understanding is established, each number may be associated the appropriate point.
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[On teaching place value, first teach ten and multiples of ten, then teach other two-digit numerals. Children may have difficulty with twenty, thirty, and fifty because those don't sound much like the related one-digit numerals -- two, three, and five.]
Preliminary work in naming ten and multiples of ten should always involve the grouping of real objects. It is important that the models for this purpose illustrate the idea clearly. A variety of models for ten, some to demonstrate numerousness and some to illustrate unity, should be used so that children do not identify ideas with only one object or device. Later children can use those concrete devices which depend strictly on position for interpretation of value.
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The operation of addition, subtraction, multiplication, and division... are, in a sense shortcuts to counting.
[This is why we learn the operations -- otherwise, dealing with very large numbers would be extremely difficult or impossible. Somewhere I read that when Ancient Greek generals needed to count their soldiers, they set up a corral of a specific size and had the companies of soldiers pack themselves into it. The size was designed to hold 10,000 men, and that batch counted as "one myriad." This would be done as many times as needed until the whole army had been counted -- or rather, measured!]
When one works with young children, the basic facts related to an operatin should first be developed with reference to actions on sets of real objects. The objects used should be alike in form and color so that the children are not confused by qualitative differences and can concentrate more easily on the essential attribute of number.